Valve tappets of the pre-cited type are used in valve gears of internal combustion engines and serve to transmit a lifting motion of a control cam to a gas exchange valve. It is known in this connection, for example, from U.S. Pat. No. 4,909,198, to arrange both upper and lower contact discs in the region of the bottom of the tappet. In this document, which is considered as species-defining, the lower contact disc serves as a wear protection disc because the tappet housing as a whole is made of a light weight material such as aluminium. The upper contact disc of this prior art document, which has a substantially larger diameter than the lower contact disc, serves not only for an initial adjustment of valve clearance during assembly of the internal combustion engine but also for an adjustment of valve clearance in accordance to the operating periods of the internal combustion engine proposed by the manufacturer. During an initial mounting of the cup tappet already, adjusting ranges of, for instance, up to 3.5 mm fluctuations in thickness have to be realized. Thus, the force of the valve spring which loads the cup tappet towards the cam has to match the highest conceivable mass of the tappet and is thus higher than desirable. These possibly, relatively high oscillating masses of the tappet cause an unnecessary increase of friction work in the valve gear. In the case of subsequent adjustment of valve clearance during maintenance work on the internal combustion engine, the valve clearance is likewise corrected via the upper contact disc which therefore has a solid configuration. This means that upper contact discs in a large number of different thicknesses have to be kept in store by engine manufacturers and servicing stations. The main drawback of the prior art tappet of U.S. Pat. No. 4,909,198, however, is that it has a relatively large mass despite its light weight housing because valve clearance has to be compensated through its upper contact disc which is relatively "solid" compared to the lower contact disc and because it has a large adjusting range.